The LQR (linear quadratic regulator) control problem subject to singular system constitutes a optimization problem in which one must be find an optimal control that satisfy the singular system and simultaneously to optimize the quadratic objective functional. In this paper we establish a sufficient condition to obtain the optimal control of discounted LQR optimization problem subject to disturbanced singular system where the disturbance is time varying. The considered problem is solved by transforming the discounted LQR control problem subject to disturbanced singular system into the normal LQR control problem. Some available results in literatures of the normal LQR control problem be used to find the sufficient conditions for the existence of the optimal control for discounted LQR control problem subject to disturbanced singular system. The final result of this paper is in the form a method to find the optimal control of discounted LQR optimization problem subject to disturbanced singular system. The result shows that the disturbance is vanish with the passage of time.

This paper presents an adaptive particle swarm optimization (APSO) based LQR controller for optimal tuning of state feedback controller gains for a class of under actuated system (Inverted pendulum). Normally, the weights of LQR controller are chosen based on trial and error approach to obtain the optimum controller gains, but it is often cumbersome and tedious to tune the controller gains via trial and error method. To address this problem, an intelligent approach employing adaptive PSO (APSO) for optimum tuning of LQR is proposed. In this approach, an adaptive inertia weight factor (AIWF), which adjusts the inertia weight according to the success rate of the particles, is employed to not only speed up the search process but also to increase the accuracy of the algorithm towards obtaining the optimum controller gain. The performance of the proposed approach is tested on a bench mark inverted pendulum system, and the experimental results of APSO are compared with that of the conventional PSO and GA. Experimental results prove that the proposed algorithm remarkably improves the convergence speed and precision of PSO in obtaining the robust trajectory tracking of inverted pendulum.

The underframe passive inerter-based suspended device, based on the inerter-spring-damper vibration attenuation structure, could improve the dynamic performance of the train body, but its parameters are fixed and cannot meet the dynamic performance requirements under different operating conditions. Therefore, a semi-active inerter-based suspended device based on the linear quadratic regulator (LQR) control strategy is proposed to further enhance the dynamic performance. The rigid-flexible coupling vertical dynamic model of the train body and an underframe semi-active inerter-based suspended device are established. The structural parameters of the semi-active inerter-based suspended device are adjusted using LQR control strategy. Dynamic response of the system is obtained using the virtual excitation method. The dynamic characteristic of the system is evaluated using the Sperling index and compared with those of the passive and semi-active traditional suspended devices as well as the passive inerter-based suspended devices. The vertical vibration acceleration of the train body and Sperling index using the semi-active inerter-based suspended device is the smallest among the four suspended devices, which denotes the advantages of using the inerter and LQR control strategy. The semi-active inerter-based suspended device could decrease the vertical vibration acceleration of the train body and further suppress its elastic vibration in the lower frequency band, more effectively than the other three suspended devices. Overall, the semi-active inerter-based suspended device could significantly reduce elastic vibration of the train body and improve its dynamical performance.